Coupled nonlinear waves in two-dimensional lattice

For one-dimensional nonlinear lattices, such as Toda lattice, it has been extensively studied. By considering the nonlinear effects of two-dimensional lattice, we set up the equation of motion for each particles (atoms, molecules or ions). For small amplitude and long wavelength nonlinear waves in this system, both the linear dispersion relation and the coupled Korteweg de Vries (KdV) equation are obtained. The simple soliton solution is obtained. If the nonlinear lattice is symmetric in the x and y directions, It is noted that there are two kinds of solitons. one is that propagates in either x or y directions, (1, 0) or (0, 1), the other is that propagates in the direction of (1, 1). It is in agreements with that of one-dimensional lattice. The different properties are investigated for different nonlinear interacting potentials, such as Toda potential, Morse potential and LJ potential.